Rheological Characterization and Modeling of Laponite Gels
with Application to Slug-like Locomotion
by
Douglas C. Hwang
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Bachelor of Science
at the
MASSACHUSET S INTT
OF TECHNOLOGY
Massachusetts Institute of Technology
June 2005
JUN 08 205
LIBRARIES
© 2005 Massachusetts Institute of Technology
All rights reserved
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Signatureof Author...........................................
4 ................
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/
Depar nent of MechanicalEngineerjg
a
vMay
6, 2005
C ertified by ...........................................................
, j~Gar
'Professor of Mechar
kI. McKinley
l Engineering
-"~~~~~~
~Thesis Supervisor
Accepted by ................................ .........................
...
Professor Ernest G. Cravalho
Chairman of the Undergraduate Thesis Committee
'.
CHIVES
1
2
Rheological Characterization and Modeling of Laponite Gels
with Application to Slug-like Locomotion
by
Douglas C. Hwang
Submitted to the Department of Mechanical Engineering
on May 6, 2005 in Partial Fulfillment of the Requirements
for the Degree of Bachelor of Science in
Mechanical Engineering
ABSTRACT
Using various concentrations of Laponite, results were obtained on an AR1000 rheometer
through various testing methods: stress sweep, creep tests, and large amplitude oscillatory
shear tests, defined as LAOS. Concentrations of 1%, 2%, 2.5%, 3%, and 4% wt. of
Laponite and distilled water solutions were tested. Laponite gels over 1% wt.
concentrations were characterized as yield stress materials, as determined by stress
sweep, creep test, and LAOS tests. The stress sweeps determined the yield stress, and the
creep tests verified the results with a range of creep tests over and below the yield stress
discovered in the stress sweep tests. The LAOS tests mapped a specific "fingerprint" of
how the Laponite gel behaved on a Lissajous figure of stress and strain. These LAOS
results are then fit with an evolution of deformation model in Matlab over various
oscillation stresses. The results show that slug slime emulation is possible by altering the
Laponite gel's properties with polymers to result in a slug slime equivalent for use in
studying slug locomotion.
Thesis Supervisor: Gareth H. McKinley
Title: Professor of Mechanical Engineering
3
4
Acknowledgements
First of all, I owe a lot to Professor McKinley who has personally taught me the past 3
years at MIT, from 2.006 Thermal Fluids, to a UROP junior year, to my senior thesis.
Thank to you for your patience and guidance for I was able to obtain a true educational
experience at MIT. Thank you for taking as one of your students, personally investing so
much time into me, being patient with me, and bringing out more potential in me than
any other professor at MIT.
Suraj Deshmukh for your help in the lab, teaching me to work with all the equipment,
working with me to understand my data, and teaching me the analysis needed in order to
complete my thesis.
For all the memories and support the past four years, I want to give thanks to my
housemates at Phi Delta Theta. And also I'd like to give recognition to all my fellow
course 2 buds, whom I have shared so many suffering as well as rewarding moments
with.
Thank you, Pastor Paul Becky JDSN, and the family of God at Berkland Baptist Church,
thank you for your countless prayers and support over my college years at MIT. David
JDSN, Angela SMN, and James hyung thank you for being such loving and caring
shepherds to me, and being spiritual parents to me throughout college. As difficult as I
have been to deal with, I have been the recipient for so much love, prayer, and sacrifice.
ABSK staff, thank you for being older brothers and sisters in Christ and being living
examples to me, teaching me to grow together in the family of God.
I want to thank my family, parents Hosuk and Eunsoo Hwang, and sister Jennifer Hwang
for all their love and support throughout my college days and entire life. Thank you for
sacrificing so much that I could come so far and finish MIT. Not of anything that I did,
but instead all this is truly a testament of the abounding love that I have received all
throughout my life.
Last, I would really like to thank Jesus Christ for being the Lord and savior of my life.
Though I am thankless, undeserving, and a sinner that I've been the recipient of so much
unconditional love. And most of all, for my salvation that I could have my sins forgiven
so that I can have a eternal relationship with God.
But he said to me, "My grace is sufficientfor you, for my power is made perfect in
weakness. " Therefore I will boast all the more gladly about my weaknesses, so that
Christ's power may rest on me... For when I am weak, then I am strong.
2 Corinthians 12:9-10
5
6
Table of Contents
1.0 Introduction ................................................................................................................. 19
1.1 Motivation............................................................................................................... 19
1.2 Goals .......................................................................................................................
20
1.3 Overview................................................................................................................. 20
2.0 Background ................................................................................................................. 22
2.1 Yield Stress .............................................................................................................
22
2.2 Material...................................................................................................................25
2.2.1 Structure ...........................................................................................................
25
2.2.2 Current Uses..................................................................................................... 26
2.3 Slug Locomotion..................................................................................................... 27
3.0 Theoretical Analysis ................................................................................................... 31
3.1 Stress Sweep and Observing Yield Stress ..............................................................
31
3.2 Creep mand
Observing Yield Stress ..........................................................................
34
3.3 Large Amplitude Oscillatory Shear and Observing Yield Stress.......................... 36
4.0 Experimental Methods ................................................................................................ 39
4.1 Sample Preparation .................................................................................................
40
4.2 Stress Sweep ...........................................................................................................
41
4.3 Creep ........................................................................................................................
43
4.4 Large Amplitude Oscillatory Shear ........................................................................
44
5.0 Results and Discussions .............................................................................................. 45
7
5.1 Stress Sweep ...........................................................................................................
45
5.2 Creep ........................................................................................................................
54
5.3 Large Amplitude Oscillatory Shear (LAOS) Tests .................................................
62
6.0 Conclusion.................................................................................................................. 78
References.........................................................................................................................81
Appendix: Matlab Code Used to Fit LAOS Results on Lissajous Figures...................... 84
8
9
List of Figures
Figure 1: Compares the three most used yield stress models: Bingham, Casson, and
Herschel - Burkely [4].............................................................................................. 24
Figure 2: A three dimensional structural form of Laponite which shows the electrostatic
attraction forces on the faces and edges of the surrounding Laponite crystals [7]... 26
Figure 3: Shows the size magnitude difference of Laponite compared to other colloidal
suspensions, bentonite and hectorite [7]................................................................... 26
Figure 4: Illustration of how a slug contracts and expands its muscles in order to use the
slug slime as a means of locomotion........................................................................28
Figure 5: Rough diagram of a slug fixed onto a vertical wall on a layer of slug slime.... 28
Figure 6: Theoretical hysteresis loop for a metal plotted on stress and strain axes .......... 32
Figure 7: Hysteresis curve of a sample fluid as the stress is swept up and down ............. 33
Figure 8: Matlab graph illustrating the effect of s =
Figure 9: Corniparesthe Lissajous figures of s = r/y
s = co/cr >
/ry
on creep tests ...................... 34
< 1, s = cro/cry = 1, and
using the evolution of deformation model........................................ 38
Figure 10: AR1000 used for all the stress sweep, creep, and large amplitude oscillatory
shear tests on the Laponite solutions........................................................................39
Figure 11: A sample of 3% wt. Laponite solution upside down before being shaken, the
stir bar suspended in the gel shows the elastic gel-like properties ............................ 41
Figure 12: A sample of 3% wt. Laponite solution, upside down after being shaken,
showing the viscous fluid-like properties.................................................................41
10
Figure 13: Two tests of stepped stress sweeps of a 1% wt. Laponite-water solution,
showing no yield stress characteristics.....................................................................46
Figure 14: Stress sweep test of silicone oil that showing characteristics of a Newtonian
fluid ............................................................................................................................46
Figure 15: Numerous stress sweep tests stepped up and down on 2% wt. concentration of
Laponite and water, showing yield stress characteristics and a hysteresis ............... 47
Figure 16: Numerous stress sweep tests stepped up and down on 2.5% wt. concentration
of Laponite and water, showing yield stress characteristics and a hysteresis........... 48
Figure 17: Numerous stress sweep tests stepped up and down on 3% wt. concentration of
Laponite and water, showing yield stress characteristics and a hysteresis ............... 49
Figure 18: Numerous stress sweep tests stepped up and down on 4% wt. concentration of
Laponite and water, showing yield stress characteristics and a hysteresis ............... 49
Figure 19: Semi-log scale of the 2% Laponite solution when applied a stepped stress
sweep test.................................................................................................................. 51
Figure 20: Semi-log scale of the 3% Laponite solution when applied a stepped stress
sweep test.................................................................................................................. 51
Figure 21: Semi-log scale of the 4% Laponite solution when applied a stepped stress
sweep test.................................................................................................................. 52
Figure 22: Compendium of the stepped stress sweep curves for the 1%, 2%, 2.5%, 3%
and 4% wt. Laponite solutions all at 20°C ...............................................................
53
Figure 23: A comparison of the Compliance J(t) for the creep tests on a 2% wt. solution
of Laponite when applied shear stresses from 1 Pa to 27 Pa, showing the difference
11
in creep tests done with shear stresses below the yield stress and shear stresses above
the yield stress value ................................................................................................. 54
Figure 24: A comparison of the Compliance J(t) for the creep tests on a 2.5% wt. solution
of Laponite when applied shear stresses from 2.374 Pa to 23.57 Pa, showing the
difference in creep tests with shear stresses below the yield stress value and shear
stresses above the yield stress value .........................................................................
55
Figure 25: A comparison of the Compliance J(t) for the creep tests on a 3% wt. solution
of Laponite when applied shear stresses from 15 Pa to 75 Pa, showing the difference
in creep tests with shear stresses below the yield stress value and shear stresses
above the yield stress value .......................................................................................
56
Figure 26: A comparison of the Compliance J(t) for the creep tests on a 2.5% wt. solution
of Laponite when applied shear stresses from 23.57 Pa to 120 Pa, showing the
difference in creep tests with shear stresses below the yield stress value and shear
stresses above the yield stress value .........................................................................
56
Figure 27: A fit of the creep test extracted viscosity values for a 2% wt. Laponite solution
mapped onto the stepped stress sweep data for the same 2% wt. Laponite sample.. 58
Figure 28: A fit of the creep extracted viscosity values for a 2.5% wt. Laponite solution
mapped onto the stress sweep data for the same 2.5% wt. Laponite sample ............ 60
Figure 29: Compendium of all the yield stress points gathered from the stress sweep tests
and creep tests for all the 2%, 2.5%, 3%, and 4% wt. solutions of Laponite and also
shown is the equation of a fitted power law line and a exponential line
corresponding to the yield stresses at a certain Laponite weight concentrations ...... 61
12
Figure 30: This perfect circle on a Lissajous figure representing a pure Newtonian fluid.
.............I......................................................................................................................
63
Figure 31: A Lissajous figure of stress and strain of a Newtonian fluid, silicone oil, at
input stress amplitudes of 3.761 Pa and 0.5919 Pa, showing that a Newtonian fluid is
characterized by a stress input and strain response that is close to a 90° phase
difference, showing a fingerprint close to a circle ....................................................
63
Figure 32: Experimental LAOS data on the silicone oil showing the relationship between
the phase degree difference and the oscillation stress ...............................................
64
Figure 33: Lissajous figure corresponding to a perfect elastic solid ................................
65
Figure 34: Lissajous figures of 2% wt. Laponite at s = 0.5963 and s = 0.9454, fit with
the model in the linear visco-elastic regime .............................................................
66
Figure 35: Lissajous figure of 2% wt. Laponite at s = 1.5 fit with the model as the
oscillation stress exceeds the yield stress of the solution ......................................... 67
Figure 36: Lissajous figures of 2.5% wt. Laponite at s = 0.5 and s = 0.75, fit with the
model in the linear visco-elastic regime resulting in ellipsoid Lissajous figures ..... 68
Figure 37: Lissajous figure of 2.5% wt. Laponite at s = 1.2 fit with the model as the
oscillation stress exceeds the yield stress of the 2.5% wt. Laponite solution ........... 69
Figure 38: 3% wt. Laponite solution tested at s = 0.4 and s = 0.7, showing the
progression from an ellipsoid shape to a non-linear shape that deforms the ends of
the ellipsoid...............................................................................................................70
Figure 39: Lissajous figure with both the experimental data and model fit for 3% wt.
Laponite as stress was increasing closer to yield stress, showing a nonlinear portion
of the fluid at s = 0.7................................................................................................. 71
13
Figure 40: LAOS tests of 4% wt. Laponite solution s = 0.3, s = 0.5, and s = 0.8 showing
the transition from a linear ellipsoid to a non-linear skewed ellipsoid on a Lissajous
figure of stress and strain..........................................................................................72
Figure 41: LAOS data of the 4% wt. Laponite at s = 0.8, and fit with the same fingerprint
in the model to describe the state of deformation in the fluid on a Lissajous figure of
stress and strain.........................................................................................................73
Figure 42: Showing the transition from a linear visco-elastic ellipsoid Lissajous figure to
a parallelogram nonlinear Lissajous figure of a 4% wt. Laponite solution by LAOS
testing at s = 0.5, s = 0.8, and s = 0.95 ....................................................................
74
Figure 43: The comparison of Lissajous figures near the yield stress s = 0.95, and above
the yield stress at s = 1. fit with the model for tests on the 4% wt. Laponite ......... 74
Figure 44: Storage modulus, G, 'loss modulus, G", and delta plotted with respect to the
oscillation stress for a 4% wt. Laponite solution to see where the yielding occurred
on the solution...........................................................................................................75
Figure 45: Waveform showing no phase difference between the applied stress and strain
response on a 4% wt. Laponite solution at s < ......................................................
76
Figure 46: Waveform of a 4% wt. Laponite solution at s > 1, showing the deformation
and phase difference of the strain response to the applied stress.............................. 76
14
15
List of Tables
Table 1: Error bars associated in mixing the Laponite percentage weight solutions in
distilled water............................................................................................................40
Table 2: Testing parameters when using applying the stepped stress sweep tests on the
Laponite solutions.....................................................................................................42
Table 3: General testing parameters used when applying the creep tests on the Laponite
solutions....................................................................................................................43
Table 4: General testing parameters used when applying the large amplitude oscillatory
tests on the Laponite solutions..................................................................................44
Table 5: Compilation of the yield stresses determined from the stress sweep data using
geometric mean from the viscosity before yielding and the viscosity after the
yielding.....................................................................................................................50
Table 6: Compiled yield stresses extracted from the creep tests of 2%, 2.5%, 3%, and 4%
wt. Laponite solutions...............................................................................................57
Table 7: Viscosity values for both the creep tests and stress sweeps tests at certain shear
stresses on the 2% wt. Laponite solution..................................................................59
Table 8: Viscosity values for both the creep tests and stress sweeps tests at certain shear
stresses on the 2.5% wt. Laponite solution...............................................................60
Table 9: The values used to fit the model to the experimental results of the 2% wt.
Laponite LAOS tests.................................................................................................67
16
Table 10: Values used to fit the model to the experimental results of the 2.5% wt.
Laponite.................................................................................................................... 69
Table 11: Values used to fit the model to the nonlinear experimental results of the 3% wt.
Laponite at 37.65 Pa .................................................................................................
71
Table 12: Values used to fit the model to the experimental LAOS results of the 4% wt.
Laponite solution......................................................................................................77
Table 13: Comparing yield stress values from 2%, 2.5%, 3%, and 4% Laponite samples
from the stepped stress sweeps and creep tests, with the oscillation stresses
corresponding to the non-linear Lissajous figures from the LAOS tests and from the
model.........................................................................................................................77
Table 14: Slug slime properties........................................................................................79
17
18
Chapter
1
Introduction
The mountains flowed before the Lord, the One of Sinai, before the Lord,
the God of Israel.
Judges 5:5
1.1 Motivation
From the bible verse above, one can infer that if even the mountains flow then one can
conclude that Rheology is a study of how any material flows! Rheology is a complex
subject, more exposed in higher levels of Chemical Engineering and fluids classes. Yet
similar to materials research, studying basic material functions in static mechanics, it may
be just as important to understand fluids characteristics when studying fluid applications.
This is so that one can understand how "real" fluids might respond to stresses and strains,
as any solid material would.
This undergraduate thesis was a self study on Rheology, conducted in order to gain a
more in depth knowledge about the fundamentals of Rheology. From the fundamental
fluid concepts learned from introductory classes, this project aims to not only extend
upon that knowledge, but also to gain an understanding of how Rheology is used, and
how those concepts can be applied to current research and applications. This paper will
go over some of fundamental ideas concerning Rheology using basic rheological analysis
19
in stepped shear sweeps and creep tests, and then move onto more current research
analysis using large oscillatory amplitude shear (LAOS), plotting the stress and strain
responses on Lissajous figures, and matching it to a evolution of deformation model of
complex fluids showing yield stress characteristics [1, 2].
1.2 Goals
The goal of this undergraduate thesis is to develop a basic understanding of Rheology
through fundamental rheometer testing and analysis. Furthermore, this thesis is to learn
how to use more advanced analysis using large amplitude oscillatory shear and how it can
be used to characterize a fluid's material properties. By testing Laponite (material
explanation in section 2) using basic heological tests, applying LAOS tests, and then
fitting it to a theoretical model, a more in-depth knowledge into Rheology can be
obtained. Finally, the concepts verified by this approach will be used to emulate the
properties of slug slime, leading into a further study of a mechanical slug and
understanding slug locomotion.
1.3 Overview
Chapter two reviews essential background information for this project. Chapter two
provides a quick broad understanding of yield stress, a little bit of its history and what
current discussions there are regarding that topic. Chapter two will also discuss the
material that will be extensively studied, Laponite. It will discuss its chemical structure as
well as the current uses of the synthetic clay material. By mixing Laponite with water a
"shake-gel" can be created. The chapter will also briefly explain the locomotion of slugs
and what fluid it uses to travel.
20
Chapter three describes the theoretical analysis used in understanding the data extraction
and explains the process in which the data is applied. It will explain the three types of
testing used: stress sweep tests, creep tests, and LAOS tests, and will discuss how yield
stress characteristics are extracted from the data.
Chapter four explains the procedure used to gather data, from sample preparation of the
various concentrations of Laponite, to the various settings and tests used.
Chapter five shows and explains all the relevant data gathered for this experiment. It will
compile and compare the various results obtained. The LAOS data will be matched and
fit by the evolution of deformation model.
Finally, the conclusion will summarize the main topics discussed in this paper and the
results. Further applications for Laponite and how it can be altered to be used as a
medium for slug-like locomotion are discussed.
21
Chapter 2
Background
2.1 Yield Stress
In current rheological literature there have been numerous papers, discussions, and
arguments regarding the topic of yield stress.
Yield as a general definition is used to as a term to give in or give way to. Regarding
fluids, it is used the slightly different way. The definition from An Introduction to
Rheology states that yield stress is a "stress corresponding to the transition from elastic to
plastic deformation" [3]. Among the discussions, there are generally three ideas that try
to define yield stress. Some say there is a gradual change in a yield "region", some feel as
if no such point exists, and others feel as if there is exactly one point where the yield
occurs [4].
The three most used equations to describe yield stress liquids are the Bingham plastic
model, Casson model, and the Hershel - Bulkey model[4].They are shown below:
The simplest model and probably the most well known model is the Bingham plastic
model. Proposed by Professor Eugene C. Bingham, who was one of the pioneers who
first defined Rheology [4].
22
(2.1)
o = cry +Up . Y
where, cr is the shear stress, ry is the yield stress,
is the strain rate, and
p is the
plastic viscosity.
Another model is proposed by Casson which is a modification of the Bingham model,
W,7
= U+
(2.2)
The Herschel - Bulkey equation is a three parameter model,
c
=%y+k. ?
(2.3)
A comparison of these three models is shown in figure 1. Although the Bingham model is
the simplest i is a still a good approximation of the fluid behavior of yield stress fluids.
23
1000
d
01
Q
100
10
IA
0.001
0.01
0.1
1
10
100
1000
10000
Shear rate, ? / s
Figure 1: Compares the three most used yield stress models: Bingham, Casson, and Herschel Burkely
141.
Other opinions, from researchers, mention yield stress as an "engineering reality," that
for practical purposes it can be used as an approximation for application [4]. There has
also been discussion on the idea of an "apparent yield stress." The concept of "apparent
yield stress" is when a strict yield stress isn't assumed to be there, when a sharp transition
from a low to high viscosity fluid is present [5]. Dekee also suggests an idea of how
yield stress works, that yield stress is caused by a buildup in the fluid, in which the
applied stress just needs to break down in order to flow [6].
24
As discussed, yield stress is an important concept, but one with several differing
definitions. When relating back to the engineering, as the Webster dictionary defines it,
Engineering: The application of science and mathematics by which the properties of
matter and the sources of energy in nature are made useful to people. It is the practical
aspect of engineering that drives research towards not only academic understanding, but
also useful applications that will positively impact people. Hence, being goal centric and
focusing on a problem at hand, only then can a definition of yield stress be applied, not
specific to the sake of science but for the sake of helping people.
2.2 Material
The material used in this study is Laponite RD from Southern Clay Products, Inc. in
Gonzales, TX.
2.2.1 Structure
Laponite is a purely synthetic material, and when mixed with water it creates a type of
gel, when sheared, acts as a viscous fluid. The chemical composition is
Na+0.7[(Si8 Mg5.5 Lio.8 )O2 0(OH)4 ]° 7 [7]. The particles are held together by electrostatic
forces, and when Laponite is sheared, it creates a gel-like viscous fluid.
The Laponite particles are disc shaped crystals 0.92nm thick and 25nm in diameter. The
electrostatic frce of the particles causes the crystals to attract to the negative surfaces of
the surrounding crystals, which gives the Laponite solution its gel-like formation.
Stronger concentrations, lead to stronger attractive forces due to more Laponite crystals.
These electrostatic forces cause the Laponite structure to be a three dimensional structure
of Laponite crystals as seen in figure 2 [7].
25
Figure 2: A three dimensional structural form of Laponite which shows the electrostatic attraction
forces on the faces and edges of the surrounding Laponite crystals [7].
Laponite is very similar to colloidal suspensions such as bentonite and hectorite in
structure but is magnitudes smaller, as seen in figure 3.
Californian
hectorite
Wyoming
bentonite
Laponite 100nm
Figure 3: Shows the size magnitude difference of Laponite compared to other colloidal suspensions,
bentonite and hectorite 71.
2.2.2 Current Uses
Colloid suspensions, like Laponite, are used in a variety of applications. Currently some
of the fields where Laponite is used are paint, cleaning products, drilling, cosmetics,
food, and pharmaceuticals [8].
26
Laponite is generally used where natural clays were once used [9, 10]. Laponite is
chemically pure, consistent in structure, and is colorless and odorless. It is generally used
as a thickening agent. When used in toothpaste, it allows the material to squeeze out of
the tube, while retaining the shape when applied on the toothbrush.
One of Laponite's more extensive uses is in the drilling industry. It is often used as a
drilling fluid in heavy oil drilling machinery. As the drill bit meets the rock layer, the
Laponite is used as a drilling fluid. By altering the properties of Laponite to a slurry, it is
applied to where the drilling occurs to bring back up the debris.
Further uses can be applied when polymer is added onto the Laponite substance. When a
polymer is added to a colloidal suspension, the additional polymer alters the rheological
properties of the fluid, allowing for application specific design [8].
2.3 Slug Locomotion
Slug locomotion is an interesting phenomenon. Slugs can move over most any type of
terrain, and can even climb vertically. Though slow and perhaps messy, because they
leave a slime residue as they move, they have many advantageous qualities worth
emulating. Research is currently being done at MIT Hatsopolous Microfluids Laboratory
to understand this locomotion fully, and to emulate it with robotics, via Robosnail [ 1,
12].
Figure 4 shows a brief illustration of how the muscles work in order to move on a layer of
slug slime. The slug moves its one foot, and from the outside observer, the layer in touch
with the slime moves in a wave-like pattern down the length of the snail, slowly moving
across the terrain, and leaving behind a layer of slime.
27
", '.1 '-f, 1:'O r
-
nif.
'
-V*r r 4 !"t'j"Z1-1,
- -l
IFO
Ersi and iort t-dina n
1._ _.!.[e ~.!.:~E.,pf ~.,.
t
t
1
_.x._
_
[http://www.amobrosi.com/slugbio.html]
Figure 4: Illustration of how a slug contracts and expands its muscles in order to use the slug slime as
a means of locomotion.
A rough analysis of the fluid is done below for a slug climbing vertically on a wall
(figure 5).
Slug
Slug
Figure 5: Rough diagram of a slug fixed onto a vertical wall on a layer of slug slime.
In figure 5, a slug is stuck onto its own slime and resting on the wall. The dimensions of
the slug are
28
WxDxH = V
(2.4)
where W is the width, D is the depth, H is the height, and V is the approximated
volume. Its density is given by p, and the force, F, that's acting on the slime face due to
gravity, g, is
F=mg=(p.V).g
(2.5)
The shear stress, r, that the force is causing on that face is given by
F
(p.V).g
=W
HW
HW
HW
(2.6)
which has units of pressure.
Using the Bingham plastic model to model the slug slime, the object will start falling if
=
7Y
(2.7)
ay being the yield stress of the slug slime.
After substituting and rearranging,
=
and V = WDH,
(2.8)
then one can conclude that if
UV> pDg
(2.9)
Looking at banana slugs, the second largest among the slug family, and doing an order of
magnitude estimate, the weight is about 0.2 kg [13, 14], length is about 20cm [15], and its
29
width is approximated to about 5 cm. By this estimation, that cry has be greater than 200
Pa!
As the yield stress is very high on the fluid, it is understandable how it manages to stay
attached to the wall, but it is phenomenal to see that slugs' ability to not only stay, but to
travel along vertical walls.
At this point, an application to the slug slime will be constructed using the analysis and
rheological characterization done on the Laponite fluid, coupled with the understanding
of these properties of slug slime.
30
Chapter Three
Theoretical Analysis
3.1 Stress Sweep and Observing Yield Stress
A shear stress can be applied in a step-wise manner from a low stress to a high stress.
After a certain region or point, the material transitions from a solid-elastic material, to a
viscous fluid. By applying a stress sweep, one can observe a change in the viscosity as a
function of the stress. After a yield stress material has been sheared from a low shear
stress to a high stress, and then sheared in the opposite direction from a high stress to a
low stress, hysteresis can be observed, similar to that of solid elastic theory.
Elastic solid theory, as shown in figure 6, shows that when an elastic material is applied a
shear stress, the value of the strain goes linearly upwards corresponding to the Young's
modulus. When the stress exceeds the yield stress, the material starts to strain harden, and
when unloaded, unloads corresponding to the Young's modulus and back into its original
form.
31
0'
P
Figure 6: Theoretical hysteresis loop for a metal plotted on stress and strain axes.
Similarly as an elastic solid, when applying a stress to a yield stress fluid it can be
modeled the same way. The fluid loads according to a modulus then becomes a viscous
fluid after the yield stress. When unloaded, it slowly comes back to its original form. This
phenomenon is seen in figure 7 and is verified in section 3.3.
32
1000
100
0
10
._
00
.e
1
0.1
0.01
0.01
1
0.1
10
100
Stress [Pa]
Figure 7: Hysteresis curve of a sample fluid as the stress is swept up and down.
From the graph in figure 7, using a geometric mean from the high viscosity value and the
low viscosity value, that value scaled down to the respective stress can be categorized as
the yield stress.
7y = 4t7H
7L
(3.1)
7
where, q,, is the viscosity at yield stress, qL
is the viscosity when yielded, and 77Hq
is the
viscosity before the fluid gets to the yield point.
33
3.2 Creep and Observing Yield Stress
When a material undergoes creep testing, it is tested at one stress value and the
deformation is shown over time. It deforms the fluid and measures the strain with respect
to the length of time the stress is applied. In different ratios of input stress to yield stress
(s =
) different fluid strain responses are seen in figure 8.
/
103
102
101
'~' '
100
-7,
~ ~
/,f-
~
- -- -- -- --
- - -- -
~
~
~
~
_
-
~
101
-
-2
10
=n 7
- --------
3
I n-
102
10
-1
100
101
Figure 8: Matlab graph illustrating the effect of s = o'o/7,
2
10
3
10
4
10
on creep tests.
Values under a ratio of s<l, the material strain approaches a constant value. As the ratio
of input stress to yield stress exceeds 1, the strain values increase linearly with time on a
log scale. Citemrne[16] used the creep graphs to see a visible difference between the
results of s < 1 and s >
corresponding to visible difference in an elastic response and a
34
viscous response. By using observing those visible differences a yield stress can be
determined from the transition from the elastic response to the viscous response. This
same analysis can be used in this case to approximate the yield stress and match it with
the yield stress values obtained in the shear stress analysis.
By doing creep tests on yield stress values obtained from the stress sweeps, a range of
effects can be seen with respect to various input stresses. By applying a stress in which
exhibits high viscosity, the strain levels out at a low value, and by applying a stress which
exhibits low viscosity, the material linearly deforms on a log-log plot. By applying a
creep test with s = 1, as determined on the stress sweep, and marginally increasing and
decreasing that value for use in creep tests, one can see where the transition from an
elastic-solid to a viscous liquid occurs. These tests can verify the yield stress value
obtained in the stress sweep tests.
By plotting the strain graphs as compliance, the opposite of a modulus, one can also get a
better look at the viscosity analysis. How the viscosity values are extracted from a
compliance graph is shown in the simple analysis below.
Stress by definition is:
(3.2)
where cr is the stress, ir is the viscosity, and
is the strain rate.
This can also be shown as:
dy
=77 -
dt
where y is the strain, and t is time.
35
(3.3)
Integrating with the initial condition that at t = 0, 7 = 0, and rearranging,
'
ar
7 =-t
(3.4)
Compliance is defined as strain over stress, finally giving the relationship.
J=-y= t
(7
(3.5)
77
where J is the compliance.
Using this relationship, one can fit the viscosity values from the compliance graphs onto
a stress sweep graph to verify the obtained results.
By comparing the yield stress from the stress sweeps with an estimated one from the
creep experiments and fitting the viscosity values from the compliance graphs, one can
see if the two tests are comparable.
3.3 Large Amplitude Oscillatory Shear and
Observing Yield Stress
By applying the equations [1, 2] the coupled nonlinear equations below describe the state
of deformation in a fluid showing yield stress properties.
When an oscillatory shear stress is applied a strain rate can be modeled as:
36
Y=
IG [a,, cos (wt) -G A(t)]
(3.6)
where y is the strain rate, r is the time constant, G is the modulus, cro is the input stress,
11
is the frequency, t is the time, and A(t) is the population imbalance of defects in the
material.
The rate of change of the population imbalance of defects in the material is given by:
' G
,&=y311
- r. cos(wt)A(t)
1
2
(3.7)
cy
where
y is the yield stress.
When the coupled non-linear ordinary differential equations are solved for materials
fitted with a proper modulus, time constant, frequency, input stress, and yield stress,
solutions can then be applied on a Lissajous figure of stress and strain. This would show
a rheological "fingerprint" illustrating the characteristics of the material at certain
parameters. For instance as a function of input stress to yield stress, s = cro/ C y , various
different "fingerprints" can be examined. When s = .o /
y
< 1, the linear elastic response
is shown, resembling a straight elliptical shape in the Lissajous figure. As s = a,/C,
approaches 1. the non-linearities are introduced into the Lissajous figure and it becomes
bigger with the edges becoming more skewed. As s = cro/y exceeds 1, the Lissajous
figure becomes a parallelogram, showing the accumulated deformation and strain in the
Lissajous figure. The various Lissajous figures corresponding to different input stress to
yield stress ratios are seen in figure 9.
37
60
I/
-
,
/
I
/
40
/
I
20
/
/
P-
0c
I
0
I
/
cj
I
I
-20
I
/
I
\I
/
/
I
/
,/
/
/
11I
(
-0.4
-0.4
I
/I.
-40
A-n
/
l
--~
-
-
I
I
-0.3
-0.2
I
I
I-
----
-
I
......
I
0
Strain
-0.1
Figure 9: Compares the Lissajous figures of
=
I
I
I
I
II
0.1
0.2
0.3
CO>/G < 1, s = o/a,
S<
1
- s=1
= 1, and
1
s>1
0.4
= CO/
>
using the evolution of deformation model.
By using large amplitude oscillatory shear, the Lissajous figures can be used to further
verify the yield stress values from the stress sweep tests and creep tests, by exhibiting
specific shapes corresponding to the values from the stress sweep and creep tests.
38
Chapter Four
Experimental Methods
The experiments performed were three different types of tests to characterize the
concentrations Laponite gels: stepped stress sweeps, creep tests, and large amplitude
oscillatory tests. All tests were performed on an AR1 000 rheometer, seen in figure 10,
with a 40mm. 1.59 degree cone and plate geometry all at 20 ° C.
Figure 10: AR1000 used for all the stress sweep, creep, and large amplitude oscillatory shear tests on
the Laponite solutions.
39
4.1 Sample Preparation
Samples were prepared by mixing Laponite with distilled water. The weight percentages
that were prepared were 1%, 2%, 2.5%, 3%, and 4% wt. solutions. The mixture was
prepared at room temperature into 100 mL solutions, and then mixed for 72 hours to
ensure complete mixture. Then, the solutions then sat at room temperature for 3 days to
make sure the Laponite mixture settled and the beginning aging effects would not affect
the results significantly [17].
The fully mixed solutions were mixed were carefully placed on a scale. The Laponite was
measured to 0.00Olgaccuracy and the distilled water was measured to 0.Olg accuracy.
The error bars are noted in the table below.
Table 1: Error bars associated in mixing the Laponite percentage weight solutions in distilled water.
Concentration
Error Bars
1%
+ 0.1005
± 0.0412
2%
2.5%
3%
4%
± 0.0412
± 0.0348
± 0.0264
The 2%, 2.5%O,3%, and 4% Laponite distilled water mixtures that were made a type of
gel material. The solutions when left out undisturbed would feel like a gel type material.
When shaken, the solution would then flow like a viscous fluid. Figures 12-13, show a
graphic illustration of the Laponite mixture states, before being shaken and after being
shaken.
40
Figure 11: A sample of 3% wt. Laponite solution
Figure 12: A sample of 3% wt. Laponite solution,
upside down before being shaken, the stir bar
upside down after being shaken, showing the
suspended in the gel shows the elastic gel-like
viscous fluid-like properties.
properties.
4.2 Stress Sweep
By applying a stepped stress sweep on each of the concentrations, a relationship of stress
to viscosity was shown. The experimental parameters are shown in table 2 below.
41
Table 2: Testing parameters when using applying the stepped stress sweep tests on the Laponite
solutions.
Conditioning Step:
Initial Temperature
Pre-Shear
Equilibrium
20 C
10 seconds
10 seconds
Steady Step Flow Step:
Stress Range [Pa]
5.968 E-3
5968
Steady State:
Percentage Tolerance
5;%
Points a Decade
Temperature
Sample Period
5
20 C
20 seconds
Consecutive within
Tolerance
3
Max. point time
1 minute
The observations during testing show that the sample does not reach the equilibrium
stated in the range specified by the 5% tolerance, but it would at least stay within the 10%
tolerance range.
The stress sweep tests were stepped up from 0.005968 Pa to 5968 Pa. On the step up
procedure, the test would finish when the rheometer would overspeed. Once the
procedure was complete, on the same sample, the stress would then be stepped back
down from the last point 0.005968 Pa. The rheometer would then stop gathering data at
the point where the change was too small for the resolution of the rheometer to pick up.
This shear step up and down procedure made the hysteresis in the fluid visible.
The main goal was to understand the yield stress of the material at certain concentrations.
Creep tests would directly follow the stress sweep tests to verify the data received.
42
4.3 Creep
By applying a creep test we were able to see at certain stresses how the strain would
change over time. Testing was done at various stresses, ranging from the solid-like
regime to the liquid-like regime in order to see the stark contrast between the two. By
observing the results in the stress sweeps, values were chosen for the solid regime, and
the viscous regime, depending on the high and low viscosities the shear stresses would
correlate to. For each fluid, thorough testing was performed to verify the yield stress
results for the stress sweep step tests, by testing points to see clearly what the yield stress
might be. The testing parameters are shown below.
Table 3: General testing parameters used when applying the creep tests on the Laponite solutions.
Conditioning Step:
Initial Temperature
Pre-Shear
Equilibrium
20 C
10 seconds
10 seconds
Temperature
Equilibrium Time
Points a Decade
20 C
2 minutes
5
Percentage Tolerance
Consecutive within
Tolerance
Sample Period
5%
3
20 seconds
Retardation:
Steady State:
43
4.4 Large Amplitude Oscillatory Shear
For each Laponite concentration an oscillatory shear was applied. For a stepped
oscillation stress input a strain response was obtained. The tests were performed at 0.1
Hz, pre-sheared, and at a sampling cycle of 5 cycles. Once obtained, the data was then
compared to the visco-elasto-plastic
model in section 3.3. The stress oscillation gave an
overview of the points and showed how it compared to the stress sweep results. A strain
sweep was also done in order to get more data close to where the ratio of input stress
approached the yield stress. An AR2000 was used for the strain sweep tests since the
AR2000 has a better resolution than the AR1000.
Table 4: General testing parameters used when applying the large amplitude oscillatory tests on the
Laponite solutions.
Conditioning Step:
Initial Temperature
Pre-Shear
Equilibrium
20 C
10 seconds
10 seconds
Steady Sweep Step:
Sweep
[micro N m
Points a Decade
Temperature
Equilibrium
Time
Frequency [Hz]
5
20 C
1 minute
0.1
0.10 - L.OOE5
Steady State:
Percentage Tolerance
Consecutive within
Tolerance
Max. Point Time
5%
3
1 minute
44
Chapter 5
Results and Discussions
5.1 Stress Sweep
Using the stepped stress sweep methods described in section 4, tests were applied to 1%,
2%, 2.5%, 3%/, and 4% wt. solutions of Laponite.
Figure 13 shows the stepped stress sweep data for the 1% wt Laponite solution. It shows
that it is very similar to that of a Newtonian liquid. It is evident it does not show a yield
stress, or a characteristic that at a certain stresses there is a difference in how the material
flows. The comparison to a Newtonian fluid can be seen in figure 14. The plot in figure
14 shows silicone oil tested at the same testing procedure as the 1% wt. Laponite. It
shows a low viscosity and a constant viscosity value throughout the entire stepped stress
sweep. Understanding there are no interesting characteristics for the % wt. solution, no
further testing was applied to that sample.
45
1.000
0.100
I
En
0
0.010
e, noda
ii Illll
0.01
10
0.1
Shear Stress [Pa]
Figure 13: Two tests of stepped stress sweeps of a 1% wt. Laponite-water solution, showing no yield
stress characteristics.
100
10
(U
0
1
.0
0.1
0.01
0.001
0.01
0.1
1
10
100
1000
Shear Stress [Pa]
Figure 14: Stress sweep test of silicone oil that showing characteristics of a Newtonian fluid.
46
Figure 15 shows a 2% wt. concentration of the Laponite water mixture. This solution
shows very different characteristics than that of the 1% wt. concentration. Through a
stepped stress sweep up and down, a hysteresis can be shown as mentioned in section 4.1.
In the region of 1Pa to 1OPa, there is a slow drop in viscosity, corresponding to the fluid
yielding in that region.
By applying a simple geometric mean, using equation 3.1, of the viscosity before yielding
and viscosity after yielding, the corresponding yield stress is 4 Pa.
10000
1000
100
in
10
0u
so
0.1
rn ti1
0.0(01
0.01
0.1
1
10
100
Stress [Pa]
Figure 15: Numerous stress sweep tests stepped up and down on 2% wt. concentration of Laponite
and water, showing yield stress characteristics and a hysteresis.
The 2.5% wt. concentration of Laponite in figure 16 shows a steeper decrease in viscosity
than shown in the 2% wt sample. This demonstrates a more defined yield stress as
opposed to a region of yielding in the 2% wt. concentration in figure 2.
47
1.00E+05
1.OOE+04
1.00E+03
'.
1.00E+02
._
0
U 1.00E+01
1.00E+00
1.OOE-01
I nnlF-n?
0.01
0.1
1
10
100
Shear Stress [Pa]
Figure 16: Numerous stress sweep tests stepped up and down on 2.5% wt. concentration of Laponite
and water, showing yield stress characteristics and a hysteresis.
The following figures 17-18 show that the 3% wt. concentration and the 4%
concentrations have an even steeper drop in viscosity after a certain yield stress or point.
Another observation shows that the hysteresis the reverse stress sweep goes back to the
original viscosity much more quickly in the higher concentrations than the lower
concentrations, showing a faster restructuring time than the lower concentrations.
48
4 fnt.A
I III Ir~l
I~
_
1.00E+05
1.00E+04
U' 1.00E+03
.'0 1.00E+02
s0
o
(0
> 1.00E+01
1.00E+00
1.OOE-01
.. -11 ivun,
I-z
1
10
100
Shear Stress[Pa]
Figure 17: Numerous stress sweep tests stepped up and down on 3% wt. concentration of Laponite
and water, showing yield stress characteristics and a hysteresis.
1.OOE+07
1.00E+06
1.OOE+05
1 OOE+04
xI
an
1.DOE+03
P_
1.00E+02
._
1.OOE+01
1.OOE+00
1.OOE-01
1.OOE-02
10
100
1000
ShearStress[Pal
Figure 18: Numerous stress sweep tests stepped up and down on 4% wt. concentration of Laponite
and water, showing yield stress characteristics and a hysteresis.
49
By combining the above stepped stress sweep graphs, a clear difference can be seen in
the various concentrations. In figure 13, the 1% wt. Laponite solution shows no yield
stress. The 2%/, 2.5%, 3%, and 4% wt. Laponite all show clear signs of yield stress. The
only exception might be the 2% wt. Laponite solution, it is a very gradual perhaps
showing a yield stress region. By applying the geometric mean, table 5 shows the yield
stresses correlating with the stress sweep tests.
Table 5: Compilation of the yield stresses determined from the stress sweep data using geometric
mean from the viscosity before yielding and the viscosity after the yielding.
Concentration
Yield Stress
1%
2%
2.5%
3%
4%
none
4 Pa
11 Pa
52 Pa
75 Pa
On a log-log scale it looks as if the viscosity is decreasing drastically when the shear
stress is above the yield stress. Though, when looking at the graphs on a semi-log scale,
one can observe that the viscosity of all the samples drops in about an absolute 20 Pa
range. Figures 19-21 show the stress sweep data for the 2%, 3%, and 4% solutions on a
semi-log scale.
50
10000
20° C
iIi
1000
I
100
o
4.'I
.P_t
Z)
0
10
U
U)
1
0.1
nn1v.v
5
0
20
15
10
35
30
25
40
Stress [Pa]
Figure 19: Semi-log scale of the 2% Laponite solution when applied a stepped stress sweep test.
dI "I".
iii i1~1
. . 11.1 -
Iti
- -1
I
1.00E+05
1.OOE+04
1.OOE+03
0.
1.00E+02
co
0o
o)
1.00E+01
1.OOE+00
1.OOE-01
14 rlrl=
,
r")
.
.
I
.
I
.
.
I
[J[Ir-[lO
vv-
vo
0
10
20
30
40
50
60
70
Shear Stress [Pa]
Figure 20: Semi-log scale of the 3% Laponite solution when applied a stepped stress sweep test.
51
1 .00E+07
1 .00E+06
1 .00E+05
1 .00E+04
.
1.00E+03
._
0 1.00E+02
0en
1.00E+01
1 .00E+00
1 .OOE-01
1 nI:.n9
10
30
50
70
90
110
130
Shear Stress [Pa]
Figure 21: Semi-log scale of the 4% Laponite solution when applied a stepped stress sweep test.
Though it may seem that with each increase in Laponite concentration, there is a steeper
drop in viscosity, the drop in viscosity is mainly done in a 20 Pa window in the fluid. In
figure 22, plots of all the stepped stress sweep tests are combined onto one graph. It
shows the difference in how each concentration mixture is affected by the varying shear
stresses. An increase concentration of Laponite corresponds to higher yield stress values.
52
1.OOE+07
-
20° C
1 .00E+06
1 .00E+05
1.OOE+04
(0
1 .00E+03
._
1.OOE+02
2% wt. Lapc
0U
(0
to 1.OOE+OI
._~
1.OOE+00
1 .00E-0'I
1 .00OE-02'
1.00E-03
0.01I
1% wt. Laponite
,,,,=-.
0.1
.
=,,
I , ,I
1
10
I
. . .
I .1
I
I I I I ' ll
100
Shear Stress [Pa]
Figure 22: Compendium of the stepped stress sweep curves for the 1%, 2%, 2.5%, 3% and 4% wt.
Laponite solutions all at 20°C.
53
1000
5.2 Creep
Creep testing was conducted to verify the stepped stress sweep tests. For 2% wt.
Laponite, the yield stress according to the stress sweeps were determined to be around 4
Pa. In the creep tests shown in Figure 23, one can see that there is a big difference
between the 6 Pa test and the 9 Pa tests, while the 15 Pa test and 9 Pa test behaved
similarly. For such a big difference in stress values and to behave similarly, and for such
a small difference in values to behave so differently shows that the yield stress must be
around the 6 Pa mark. This evidence can also be seen in the difference of the 6 Pa creep
test and the 3 Pa creep test. Since there is a drastic difference in the way the fluid flows,
one can assume that there be a corresponding yield stress in that region of shear stress
values.
1.OOE+04
1.OOE+03
1.OOE+02
1.OOE+01
W
1.00E+00
°
c,-
E
1.O0E-O1
1.OOE-02
0.
E
1.OOE-02
1.OOE-03
1.OOE-04
1.OOE-05
I
flF-A
1.OOE-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00OE+02
1.OOE+03
Time [s]
Figure 23: A comparison of the Compliance J(t) for the creep tests on a 2% wt. solution of Laponite
when applied shear stresses from I Pa to 27 Pa, showing the difference in creep tests done with shear
stresses below the yield stress and shear stresses above the yield stress value.
54
By applying the same analysis for the 2% wt. Laponite solution in figure 23, yield
stresses can be estimate in the creep graphs in figures 24-26 for the 2.5%, 3%, and 4%
wt. Laponite data.
d
1
nnclns
,,[,FT,,a
I- I1
I
.
UVLTVV
/
..
co
y-
1.OOE+03
1.00E+02
1.OOE+01
1.OOE+00
'U
-
1.OOE-01
1.OOE-02
E
o
(J 1.00E-03
23.57 Pa
-12 Pa
9.44 Pa
1.OOE-04
1.00E-05
1.OOE-06
I
, I 111-7
11-1
I
I
Illl~--II
1.00E-.03
I
I
1.OOE-02
.
.
, ., I
1.OOE-01
I
.
I
,, I
I
1.OOE+00
I
I , , ,, I
1.00E+01
I I , ,, I
-
6 Pa
-2.3~74
-1-1 I
I
1.00E+02
I I
__j
I
1.00E+03
Time [s]
Figure 24: A comparison of the Compliance J(t) for the creep tests on a 2.5% wt. solution of Laponite
when applied shear stresses from 2.374 Pa to 23.57 Pa, showing the difference in creep tests with
shear stresses below the yield stress value and shear stresses above the yield stress value.
55
1.OOE+04
1.00E+03
1.OOE+02
1.OOE+01
m
,.
1.00E+00
S
U
=
1.OOE-01
L
E
U
1.OOE-02
1.OOE-03
1.00E-04
1.OOE-05
I..___
nnF-6.,_
I.00OE-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
Time [s]
Figure 25: A comparison of the Compliance J(t) for the creep tests on a 3% wt. solution of Laponite
when applied shear stresses from 15 Pa to 75 Pa, showing the difference in creep tests with shear
stresses below the yield stress value and shear stresses above the yield stress value.
1.OOE+04
1.00E+03
1.00E+02
1.00E+D1
E.
1.OOE+D00k
U
C
1.OE-O:)1
E 1.0OE-302
(,)
1.00E-413
20 Pa
00 Pa
75Pa
7.23 Pa
1.00E4:14
1.00E-05
I
.P'.
1 'W
i t JlJF ~ ~t]
_
I J/, 7-
-.
_
1.OOE-03
1.00E-02
1.O00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
Time [s]
Figure 26: A comparison of the Compliance J(t) for the creep tests on a 2.5% wt. solution of Laponite
when applied shear stresses from 23.57 Pa to 120 Pa, showing the difference in creep tests with shear
stresses below the yield stress value and shear stresses above the yield stress value.
56
Compiling all the data gathered from the creep tests, table 6 shows the yield stresses
obtained from the creep analysis.
Table 6: Compiled yield stresses extracted from the creep tests of 2%, 2.5%, 3%, and 4% wt.
Laponite solutions.
Concentrations
Yield Stress
2%
2.5%
3%
4%
5 Pa
12 Pa
32 Pa
90 Pa
Now using any one of these graphs, one can also verify the viscosity results of the stress
sweep tests by using the analysis in section 3.3. Using the Compliance graph of 2% wt.
Laponite in figure 22, the viscosity values can be extracted from the graph and fitted
right onto figure 15, the stepped stress sweep plot of 2% wt. Laponite. One can see that
the overlap in figure 27 is reasonable. This allows the reassurance that the two test results
are similar with also similar yield stresses.
57
1000
Stress Sweep Results
Stress Sweep Resutls2
*
100
Creep Model Frt
10
a.
0U
(A
1
0.1
I
[
t"%
[
I
I
0.01
0.10
1.00
10.00
100.00
Stress [Pa]
Figure 27: A fit of the creep test extracted viscosity values for a 2% wt. Laponite solution mapped
onto the stepped stress sweep data for the same 2% wt. Laponite sample.
Table 7, shows the comparison of the stress sweep test viscosity values to the creep test
viscosity values to evaluate how closely the values line up.
58
Table 7: Viscosity values for both the creep tests and stress sweeps tests at certain shear stresses on
the 2% wt. Laponite solution.
Shear Stress [Pa]
Viscosity [Pa s Creep
Creep
Shear Stress [Pa]
Sweep Tests #1
23.58
14.95
9.452
5.968
3.765
0.9458
27
0.033
15
0.1
9
0.33
6
1
3
33.33
1
250
Viscosity [Pa s]
Sweep Tests #1
0.02106
0.05802
0.2422
1.71
13.93
317
Shear Stress [Pa]
Sweep Tests #2
24.92
15.79
9.993
6.309
2.512
0.9999
Viscosity [Pa s]
Sweep Tests #2
0.01921
0.05419
0.2198
1.224
38.35
171.9
Figure 28 and table 8, shows samples from the 2.5% wt. Laponite solutions tests. Similar
to the 2% solution, the figure shows that the creep viscosity values follow the stress
sweep results.
59
1.OOE+-04
1 .00E+03
1 .00E+02
I
0.r
1 .OOE+01
(0
0
U)
1 .OOE+I:)0O
1 .00OE-01
I fOfF-F2
0.1
1
10
100
Shear Stress [Pa]
Figure 28: A fit of the creep extracted viscosity values for a 2.5% wt. Laponite solution mapped onto
the stress sweep data for the same 2.5% wt. Laponite sample.
Table 8: Viscosity values for both the creep tests and stress sweeps tests at certain shear stresses on
the 2.5% wt. Laponite solution.
Shear Stress [Pal
Viscosity [Pa s Creep
Creep
23.57
0.333
12
10
9.44
6
2.374
12.5
333.33
1000
Shear Stress [Pa]
Viscosity [Pa s]
Shear Stress [Pa]
Viscosity [Pa s]
Sweep Tests #1
Sweep Tests #1
Sweep Tests #2
Sweep Tests #2
23.57
14.96
9.441
5.957
3.76
0.1288
1.327
23.07
513.7
2214
27.34
12.83
N/A
5.955
2.761
0.07412
3.822
N/A
491.8
2634
60
Figure 29 shows the results of the stress sweep tests and the creep tests superimposed
onto one graph. The trend lines show the relationship between the yield stresses on a
power law curve and on an exponential relationship with respect to Laponite wt.
concentration.
nrX
A
ii ii i
iI iUUU
l
100
IL.
I
y = 0.3049E
U
C,
U)
Un
-65
._
U,
0
1E+08x4
3444
Creep and Sweep Data:
N
Sweep Tests
*
Creep Tests
10
0
- - Power (Creep and Sweep
Data:)
is
Expon. (Creep and Sweep
Data:)
F
1
-
10%
1 %i
Laponite Wt.%
Figure 29: Compendium of all the yield stress points gathered from the stress sweep tests and creep
tests for all the 2%, 2.5%, 3%, and 4% wt. solutions of Laponite and also shown is the equation of a
fitted power law line and a exponential line corresponding to the yield stresses at a certain Laponite
weight concentrations.
61
5.3 Large Amplitude Oscillatory Shear (LAOS) Tests
After applying an oscillatory shear test and outputting the form in a Lissajous figure of
stress and strain, one is able to see a fingerprint of the characteristics of the fluid.
At an oscillatory stress at
ao = sin(wt)
(5.1)
ro = cos(wt)
(5.2)
and the strain response is
This shows a 90° phase difference between the input stress and the strain response. In
figure 30 a perfect circle denoting a pure linear Newtonian fluid. Figure 31 shows the
results from two LAOS tests of the Newtonian fluid of silicone oil. A perfect circle is not
visible, since the phase difference is tilted a couple degrees, is this seen in the phase
difference plot in figure 31. This phase difference is a result of silicon oil's similar but
imperfect Newtonian fluid properties.
62
4
*
I
+
3
4+
2
*
*
+~~
+*4
*
1
(E
a)
~~~~~~~~~~~~~~~~~~~-
0.
**
Cco -1
+
-2
4-$
+
+
-++
-3
_/1
-61
D
*t
I
-40
-20
A~~~~~LIL4
0
Strain
20
40
60
+- --
Figure 30: This perfect circle on a Lissajous figure representing a pure Newtonian fluid.
4
+ + ++ ++ *
*++
3
+
2
1
*
*
*
*
-1
*
*
.C
0
a= 3.761
= 0.5919
*
+*
-
4*
*
4
-2
I
* *
I4-
**
-3
-A
-610O
l
-40
*
*-
-.
-20
l4*.
l
0
20
*
40
4-+
60
Shear Stress [Pal
Figure 31: A Lissajous figure of stress and strain of a Newtonian fluid, silicone oil, at input stress
amplitudes of 3.761 Pa and 0.5919 Pa, showing that a Newtonian fluid is characterized by a stress
input and strain response that is close to a 90° phase difference, showing a fingerprint close to a
circle.
63
Based on the results in figure 31, the phase shift for the 0.5919 Pa sample was 89.94° and
for the 3.76a Pa test, the phase shift was 90.11° which explains the imperfect circle in
figure 31. Figure 32 shows the phase difference with respect to the oscillation stress, and
it can be seen that the data wavers around 90 ° .
99
98
+
97
96
LO
a)
C.D
0)
a)
C
95
+
94
93
92
91
_
-
-3
10
10 ' 2
90
PgC
ul~~~lllsl-
+ iii
F .~+ ~"
++
++
0llll Sllll
ulll
10
+
10d
-+
slll
lllll
10
102
10 3
Oscillation Stress [Pa]
Figure 32: Experimental LAOS data on the silicone oil showing the relationship between the phase
degree difference and the oscillation stress.
On the other hand, an elastic solid would give a profile of a straight line on a Lissajous
figure of stress and strain. Where given an oscillation stress, there would be very little
strain response since the material is a linearly elastic solid. This is seen in figure 33.
64
4
I
I
I
I
I
I
,
, , , 1~~~~~~*
3
2
+~~
.. $~~~~
+
1
(1
EL
cof
o)
-1
-2
-3
.,.-*
~- ~ ~+~~~
.,.-*
~- ~ +~~~
~
.?-
~"~~'
III
+~~~
+~~~
+~~~+~~~
*
.A*,+~~
Fs~~~
w~~wr~
f ||
||
||+
_A-
-4
-3
-2
-1
0
Strain
1
2
3
4
Figure 33: Lissajous figure corresponding to a perfect elastic solid.
When applying the large amplitude oscillatory shear to the Laponite samples at different
oscillation stresses there were many different Lissajous figures that characterized the
fluid at different stages. In figure 34, a 2% wt. Laponite solution is shown at two different
oscillatory stresses on a Lissajous plot, 5.963 Pa and 9.454 Pa, corresponding to an input
stress to yield stress ratio, s = 0.5963 and s =0. 9454. The yield stress as determined by
the LAOS test and model, not the stress sweep and creep test. Fitted with that is the
model that predicts the shape of the Lissajous figure, as explained in section 4.4. The
linear visco-elastic region is shown by the elliptical figure on a Lissajous plot.
65
II I
8
6
4
co0
.)
-2
-4
-6
-8 I
-Il-
-0.08
I
-0.06
-0.04
-0.02
I
0
Strain
0.02
0.04
0.06
0.08
Figure 34: Lissajous figures of 2% wt. Laponite at s = 0.5963 and s = 0.9454, fit with the model in
the linear visco-elastic regime.
As the oscillation stress approaches the yield stress, the Lissajous figure becomes
expanded, transforming from an ellipsoid to a parallelogram. This phenomenon is also
shown in the LAOS Lissajous fingerprint model. The fitted model with experiment data
can be seen in figure 35 at s = 1.5.
66
15
10
5
0a_
0
o
-5
-10
-15
-0
4
Strain
Figure 35: Lissajous figure of 2% wt. Laponite at s = 1.5 fit with the model as the oscillation stress
exceeds the yield stress of the solution.
When fitting the model to the experimental results there are two main factors to control, T,
the time constant of the fluid and G, the modulus. The table below shows the values used
to model the 2% wt. Laponite at specific oscillation stresses.
Table 9: The values used to fit the model to the experimental results of the 2% wt. Laponite LAOS
tests.
Input Stress
Amplitude,
os,
Yield Stress,
y
Time Constant,
Modulus, G [Pal
Frequency, w
[Pa]
T [s]
5.963
10
0.25
150
0.628
9.454
10
0.25
175
0.628
14.99
10
0.22
125
0.628
[rad/sl
[Pa]
67
When observing the 2.5% Laponite mixture the results are similar to that of the 2% wt.
Laponite solution. Figure 36-37 show the model again fit to the LAOS data obtained
from the 2.5% wt. Laponite mixture on stresses ranging from 9.454 Pa to 23.75 Pa to see
the entire range of shapes from the linear elastic region to the non-linear region.
15
10
5
.0(D
0
Q.,"
0)
n
co
-5
10
-15
-U..2
U b -U.'
U
-U.
U.Ut3
U. 1
I
U. -I
,
Strain
Figure 36: Lissajous figures of 2.5% wt. Laponite at s = 0.5 and s = 0. 75, fit with the model in the
linear visco-elastic
regime resulting in ellipsoid Lissajous figures.
68
20
15
10
00
O0
Wn
a) 5
co0
PD
CO
-5
-10
-15
-20
_9;
-8
-6
-4
-2
0
2
4
6
8
Strain
Figure 37: Lissajous figure of 2.5% wt. Laponite at s = 1.2 fit with the model as the oscillation stress
exceeds the yield stress of the 2.5% wt. Laponite solution.
The tables of values used to fit the 2.5% wt. Laponite experimental results are below in
table 10.
Table 10: Values used to fit the model to the experimental results of the 2.5% wt. Laponite.
Input Stress
Amplitude,
Yield Stress,
[Pa]
Time Constant,
r
Modulus, G [Pa]
Frequency, w
[rad/s
[s]
[Pa]
9.454
20
0.2
180
0.628
14.99
20
0.3
150
0.628
23.75
20
0.012
55
0.628
69
The 3% wt. Laponite and the 4% wt. Laponite show more of the nonlinear characteristics
of the Laponite solutions. As the stress increases and comes close to the yield stress of
the fluid, one can observe that in the Lissajous figures, the edges start to stray from the
ellipsoid shape. This shows that at those oscillation stresses, the Laponite solution is
becoming more non-linear. Figure 38, shows the LAOS results of the 3% wt. Laponite
solution at 23.75 Pa or s = 0.4, where the ellipsoid shape is still retained, and at 37.65 Pa
or s = 0. 7, where the nonlinearity can be start to be seen on the edges of the ellipsoid.
50
40
30
20
10
en
e)
ICl)
0
-10
-20
-30
1
-40
Cra
_
-;0.U1
-0.1
I
-0.05
0
.
7.65 P a
.
I
0.05
.
.
I
.
0.1
Strain
Figure 38: 3% wt. Laponite solution tested at s = 0.4 and s = 0. 7, showing the progression from an
ellipsoid shape o a non-linear shape that deforms the ends of the ellipsoid.
The evolution of deformation model also fits the non-linear regions of the Laponite fluid
as shown in figure 39. The equations in section 4.4, models the population imbalance of
70
the defects in the structured fluid and predicts the deformation in the Lissajous figure due
to the increasing stress.
40
30
20
10
o
O0
-i-
-10
tO
10
-20
-30
-4An
-0.1
-0.08
-0.06
-0.04
-0.02
0
Strain
0.02
0.04
0.06
0.08
0.1
Figure 39: Lissajous figure with both the experimental data and model fit for 3% wt. Laponite as
stress was increasing closer to yield stress, showing a nonlinear portion of the fluid at s = 0. 7.
The values used to fit the LAOS result of the nonlinear 3% wt. Laponite at 37.65 Pa are
shown in table 11.
Table 11: Values used to fit the model to the nonlinear experimental results of the 3% wt. Laponite at
37.65 Pa.
Input Stress
Amplitude, cro
Yield Stress,
y
Time Constant,
[Pal
[I
55
0.1
Modulus, G [Pa]
Frequency,
[rad/s]
[Pa]
37.65
71
500
0.628
w
In the 4% wt. Laponite solutions, one is able to see the entire transition from a linear
ellipsoid shape, to a non-linear skewed ellipsoid shape, to a deformed shape where the
input stress exceeds the yield stress of the 4% wt. Laponite solution. Figure 40, shows the
linear elasto-plastic regimes of the 4% wt. Laponite fluid at 37.65 Pa (s = 0.3), 59.68 Pa
(s = 0.52), and 94.59 Pa (s = 0.8). Figure 41 shows the 94.59 Pa oscillatory stress LAOS
data plotted on a Lissajous figure of stress and strain fit with the model to match the
deformation in the fluid.
Awn,r
I ::JU
100 -
50
U
U,
en
0
-
no
-50 37.65 Pa
-
-100 -
59.68 Pa
94.59 Pa
-
n .1
Iou
,
_l
-0.15
-
I
I
I
I
-0.1
I
I
I
I
I
I
I
I
-0.05
I
I
0
I
I
I
0.05
I
I
__II
0.1
I
II
0.15
Strain
Figure 40: LAOS tests of 4% wt. Laponite solution s = 0.3, s = 0.5, and s = 0.8 showing the transition
from a linear ellipsoid to a non-linear skewed ellipsoid on a Lissajous figure of stress and strain.
72
I.1I UU
I
I
80
60
40
20
0W
co
0
-20
-40
-60
-80
4 tH--
-IUU '
-0.2
'
-0.15
'
-0.1
'
-0.05
0
'
0.05
0.1
0.15
Strain
Figure 41: LAOS data of the 4% wt. Laponite at s = 0.8, and fit with the same fingerprint in the
model to describe the state of deformation in the fluid on a Lissajous figure of stress and strain.
Figure 42-43., combines both graphs above and adds another fit of an input stress above
the yield stress to see the evolution of the fingerprints from a perfect ellipsoid, to a
skewed ellipsoid with edges moving into the non-linear regime, and finally in the
parallelogram shape showing the large changes in deformation rate and accumulated
strain [1, 2].
73
150
100
50
(1~
0
0
I)
-50
-100
-10
-0.25
-0.2
-0.15
-0.1
-0.05
0
Strain
0.05
0.1
0.15
0.2
0.25
Figure 42: Showing the transition from a linear visco-elastic ellipsoid Lissajous figure to a
parallelogram nonlinear Lissajous figure of a 4% wt. Laponite solution by LAOS testing at s = 0.5, s
= 0.8, and s = 0.95.
œrnu
100b
100
_Wu
CZ)
Gn
C')
a)
0
-50
-100
-1 0
- 8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Strain
Figure 43: The comparison of Lissajous figures near the yield stress s = 0.95, and above the yield
stress at s = 1.1 fit with the model for tests on the 4% wt. Laponite.
74
Figure 44 shows G', the storage modulus and G", the loss modulus for oscillation stress
sweep and strain sweep tests. At around 100 Pa, one can notice a drop occurs in both G'
and G" with the increase of the phase angle difference. This area is where the fluid has
been yielded.
-1
-_1-
-I IJU.U
----
1 UUUU
o
Stress sweep step
91).00
o o
Strain sweep step
1000 -90.00
1000
-6.00
o
100.0
-in
-100.0
10.00 -50.00
0.
0 0 .0 .0
.00 . . .
. 0
-10.00
o
Q
3().00~~~~~~~~~~~~
-40.00
o
1 000
1 .000
-2
0.00
--40).00
.00
-10.00
o0
o
0.1000
-0.1000
,1 0 .00
0.01000----
U.UlUUU
-
o
.
CD
o
U.1000
o
.
D
o
[]
o
1.000
o
o
0
. . . I ..
0C}
0
10.00
0
.I
O
100.0
.
0.01000
1000
osc. stress (Pa)
Figure 44: Storage modulus, G,' loss modulus, G", and delta plotted with respect to the oscillation
stress for a 4% wt. Laponite solution to see where the yielding occurred on the solution.
Figures 45-46 shows the corresponding waveforms of stress and strain responses to s <1
and s >1 on the LAOS tests on the %4 wt. Laponite. The waveform of s < 1 shows the
two oscillations in phase, while the accumulation of deformation and change of phase
angle is clearly visible in the waveform of s >1.
75
/ I \
11X
/
/\
"I
I
/
I
~A~~
'X
/
I s
II
\\Xt-S//
/
7
'
-s _ --7 '
.N/
/
Figure 45: Waveform showing no phase
Figure 46: Waveform of a 4% wt. Laponite
difference between the applied stress and strain
solution at s > 1, showing the deformation and
response on a 4% wt. Laponite solution at s < 1.
phase difference of the strain response to the
applied stress.
The nonlinearity of the LAOS plots as the input stress approaches and exceeds the yield
stress corresponds to the energy dissipated in the deformations and accumulated strain
[16]. To compare the values used in the simulation model to the values extracted from the
4% wt. Laponite results can be seen in table 12 below.
76
Table 12: Values used to fit the model to the experimental LAOS results of the 4% wt. Laponite
solution.
Input Stress
Amplitude,
o
Yield Stress, y
Time Constant,
[Pa]
T
Modulus, G [Pal
Is]
Frequency, w
[rad/s]
[Pal
59.68
115
0.1
1110
0.628
14.99
115
0.15
1025
0.628
108.8
115
1
500
0.628
131
115
.07
800
0.628
When comparing all the yield stresses obtained in the stepped stress sweep tests and the
creep tests, with Lissajous fingerprints exhibiting non-linear characteristics
corresponding to the oscillation stress on the LAOS tests and the fitted model, one can
see in table 13, that the yield stresses values are comparable. The yield stress value from
the LAOS tests is the value closest to the yield where a clear data point was obtained.
Therefore, it is a slight higher value than the yield stresses as characterized by the stress
sweep tests or creep tests. The yield stresses on the deformation model, is one that gave a
good fit to the LAOS Lissajous figures.
Table 13: Comparing yield stress values from 2%, 2.5%, 3%, and 4% Laponite samples from the
stepped stress sweeps and creep tests, with the oscillation stresses corresponding to the non-linear
Lissajous figures from the LAOS tests and from the model.
Yield Stress
Yield Stress
Yield Stress
Yield Stress
Stress Sweep
Creep Test
LAOS Tests
Model
2%
4 Pa
5 Pa
15 Pa
1OPa
2.5%
11 Pa
12 Pa
23 Pa
20 Pa
3%
52 Pa
32 Pa
38 Pa
55 Pa
4%
75 Pa
90 Pa
109 Pa
115 Pa
Concentration
77
Chapter Six
Conclusion
The goal of this thesis was to first gain a basic understanding of the methods and analysis
used in Rheology. Studying the phenomenon of yield stress, the complexities of
Laponite, and slug locomotion, a basic scope was obtained for the project. Then by
applying stepped stress sweeps and creep tests, basic and fundamental heological data
was obtained and analyzed. The stepped stress sweeps helped to first categorize the
concentrations of Laponite was yield stress fluids or not, and the creep tests verified those
results. The stepped stress sweeps categorized the 2%, 2.5%, 3%, and 4% wt. Laponite
concentrations as fluids with yield stresses. The creep tests verified the viscosity values,
as well as the yield stresses from viewing the differences in Compliance J(t) data on
whether the fluid response was that when applied a shear stress above or below the yield
stress. Those concepts then were translated when applying large amplitude oscillatory
shear tests on to the Laponite concentrations. Through the LAOS tests, the results were
plotted onto Lissajous figures of stress and strain, and specific fingerprints of
characteristics were seen at different input stress to yield stress ratios, s = (o/
C
y )
. At
s < 1, the rheological fingerprint would resemble an ellipsoid shape, giving information
that the fluid is in the linear elasto-plastic regime. As s approached 1, the nonlinearities
would be visible with an ellipsoid shape growing outward with the ends starting to
become skewed. When the input stress exceeded the yield stress, at s >1, the Lissajous
figure would start to resemble a parallelogram to show the accumulated strain and
deformation. The LAOS results were fit to a model of the evolution deformation in the
fluid. This model matched the data well.
After characterizing the Laponite fluid concentrations and observing slug slime, an
emulation process to create a artificial slug slime is possible. Looking at the properties of
78
slug slime in table 14, one can notice that the yield stress is a bit higher than
characterized by the concentrations of Laponite so far. The viscosity after yield values for
Laponite are also magnitudes below the slug slime. The heal time, though not
qualitatively measured, quantitatively the 4% wt. solution is sufficient for that heal time
value.
Table 14: Slug slime properties.
Yield Stress
Viscosity After Yielding
Heal Time
- 5 Pa
< 1second
400Pa1- 500 Pa
From this point, with the Laponite solutions characterized, one might want to increase the
yield stress by testing higher concentrations of Laponite solutions, say a 5% wt. solution.
Then by adding various polymers, more alterations in the fluid properties can be made to
fit the viscosity after yielding values.
Laponite solutions are used in a variety of applications. Through this characterization of
Laponite in the various analyses and tests, a model to emulate slug slime is established
and by adding polymer one can eventually create a solution matching the properties of
slug slime, in order to further understand the complexities and uses of slug locomotion.
Slug locomotion is an unique type of movement and potentially has many practical
applications. As nature is the most efficient engineering design on the planet, it is only
appropriate that research can be done to understand and then emulate those designs that
may be used to help in positively affecting people.
79
80
References
1.
Guo, J., et al. Rheological Fingerprinting of Nonlinear (Yield Stress) Materials by
Large Amplitude Oscillatory Shear. in The XIVth International Congress on
Rheology. 2004. Seoul, Korea.
2.
Deshrnukh, S.S. and G. McKinley. Rheological Behavior of Magnetorheological
Suspensionsunder Shear, Creep,and LargeAmplitude OscillatoryShaer (LAOS)
flow. in The XIVth International Congress on Rheology. 2004. Seoul, Korea.
3.
Barnes, H.A., J.F. Hutton, and K. Walters, An introduction to rheology. 1989,
Amsterdam; New York: Elsevier: Distributors for the U.S. and Canada Elsevier
Science Pub. Co. ix, 199.
4.
Barnes,H.A., The yield stress - a reviewor pi alpha nu tau alpharho epsilon
iota'- everything flows? Journal of Non-Newtonian Fluid Mechanics, 1999. 81(12): p. 133-178.
5.
Papanastasiou, T.C., Flows of Materials with Yield. Journal of Rheology, 1987.
31(5): p. 385-404.
6.
Dekee, D. and C.F.C.M. Fong, A True Yield Stress. Journal of Rheology, 1993.
37(4): p. 775-776.
7.
Laponite Science. 2005, Laponite.com.
8.
Zebrowski, J., et al., Shake-gels: shear-induced gelation of laponite-PEO
mixtures. Colloids and Surfaces a-Physicochemical and Engineering Aspects,
2003. 213(2-3): p. 189-197.
81
9.
Laponite Technical Information L204/Olg. 2005, Rockwood Additives Limited.
10.
Laponite Applications. 2005, Laponite.com.
11.
Chan.,B., N.J. Balmforth, and A.E. Hosoi, Building a better snail.-Lubrication
and adhesive location. In review.
12.
Chan, B., et al. Mechanical Devices for Snail-like Locomotion. in World Congress
on Biomimetics, Artificial Muscles and Nano-Bio. 2004.
13.
Richter, K.O., Aspects of Nutrient Cycling by Ariolimax-Columbianus (Mollusca,
Arionidae) in Pacific Northwest Coniferous Forests. Pedobiologia, 1979. 19(1): p.
60-74.
14.
Rollo, C.D., Consequences of Competition on the Reproduction and Mortality of
3 Species of Terrestrial Slugs. Researches on Population Ecology, 1983. 25(1): p.
20-43..
15.
Gordon, D.G. and Western Society of Malacologists., Field guide to the slug.
Sasquatch field guide series. 1994, Seattle: Sasquatch Books. 48 p.
16.
Citerne, G.P., P.J. Carreau, and M. Moan, Rheologicalproperties
ofpeanut
butter. Rheologica Acta, 2001. 40(1): p. 86-96.
17.
Mourchid, A., et al., Phase-Diagram of Colloidal Dispersions ofAnisotropic
Charged-Particles- EquilibriumProperties,Structure,and Rheologyof Laponite
Suspensions. Langmuir, 1995. 11(6): p. 1942-1950.
18.
Denny, M.W. and J.M. Gosline, The Physical-Properties of the Pedal Mucus of
the Terrestrial Slug, Ariolimax-Columbianus. Journal of Experimental Biology,
1980. 88(OCT): p. 375-393.
82
19.
Denny, M.W., Mechanical-Properties of Pedal Mucus and Their Consequences
for Gastropod Structure and Performance. American Zoologist, 1984. 24(1): p.
23-36.
83
Appendix: Matlab Code Used to Fit LAOS
Results on Lissajous Figures
1st Matlab script. Calls on the second script "equation.m" to solve the coupled non-linear
ordinary equations for the evolution of the deformation of a yield stress characteristic
fluid and outputs a Lissajous figure of stress and strain.
2s
1 ,
a..... l.g.:s }h,~s',^ng
it
a1
* 2
er
' i
as i
1 E 1 i
i
' j
_
cr
L.elj.g>rci-:N.'.a.e T;-esi
Advsz
E.-fesso:s_
Gare_;-thMc/<K-inley
Sc.5il
-:' ir-..
5
ig.
iios
clap~l S)
S' Ci
s5J.r.
t
'd'if .eOn eri._i
{ r
1
b.:!l
.Sii
c-I1
ea i
t
e
stre.ss
and strai
model.
C iissd
f- solve.
SA
t-i-heYtL-it r
Mc:and sr-]e1 t.e :or'-
s
sigma knot omega
0-.'riLv:t:Va:.
:ia i e.es
omega = 0.628;
t;freqe
G = 150;
tau = 1;
=
onstn
U
stS
arr'
v:i''Ist::e [c'
C2 = 2*pi/omega; '
ti
sR.
[Ea]
litcde
a
e--'.;'.. period
l
f"
" o'.-I) :r
t
rl
e .
aeqa
:
":-.-i;.
i ODE
C
;c
a'---arr',
ti s
fis
.s
f prids
e..rre
[0 T1];
s
"tefr.lat i.on. 'ri,~ f
[tout, yout] = ode45( 'e-t t '
- kse 'tub,. $'r:.-
e sI/
[a.
i
sigma knot = 95;
sigma_y = 115;
y
Tspan
[
%oulixs
-- C-~,d-i&,~,~Di
n
4;
Ti = n*C2;
rS
r~.ea:: 'c:,: ei
o1s.
global tau G sigmay
- :i..t.i..
_
c
Y0) = [0 0];
r
,'r_'
itrval
t-,l-.
o: i:i tecrati.
o, sc:..., ',':tS1z ©D;S
.he
T.Tspan,YO);
· ::::i:.-r: D'efricns
stress = sigmaknot*sin(omega*tout);
strain = yut(:,1);
delta = yout(:,2);
84
or
-sot c-:re.-t
I,.
on
t}-e Ii ssa-jous
....
I..
offset = (ax(strain)+min(strain))/2;
strain = strain - offset;
'-l.ots-of
:.r-ss,
s r..J-in
,a nd
]o
e]..toa
wit h
:espe-c
tm re
figure(1), plot(tout,stress), title('stress vs
':-.t)xlabel
(tim )ylabel
(1s - st)
figure(2), plot(tout,strain), title('st ai- t
label
('tioe),
ylabel('star')
figure(3), plot (tout,delta), title('el a v.
-out)t
er
x
ylabel
tout'),xlabel('tie.),
('delt-a')
Plots . LiSsajos
figurie
of stress and strin
figure(l), plot(strain,stress),
title('St:ress
s
Sza.o' ),xabel
('Strain' ) ,ylabel
('Stress Pa')
2
Matlab script used in the first script as a function which is called on to solve the
nd
ODEs.
i~'"
, :~
i-t (:i , . so
]. , ' t e J.int
'.
ti i equa .io.ns of
i:.- akes two paramretes,
ard nthe^
t a. co
coo.iti-,
....
,s span
s a.nte.
i..nntia.
n. tors
functiondydt = eqnc(t,y)
, z-l
. .1 . i I -.
t
.S
gLobal tau G sigmay
.-t.'.i'~
nt i.: r-i
fi t!_i ia-.
e = y(1);
delta
sigmaknot
omega C2
i..'i-ia ti'z
= y(2);
sigma = sigma_knot*sin(omega*t);
-fi-n
.
dgammadt
ddeltadt
. .al-.
E FuaiatiJ.
-s
of
t:.:a.-..
and Delta
= (1/(tau*G))*(sigma - G*delta);
= dgammadt * (1 - ((G*sigma*delta)/(sigmay^2)));
t-.tt [,g..atr
;
'a
dvdt = [dgammadt; ddeltadt];
85
i.n i ric
i ... a
86